High-electric-field limit for the Vlasov–Maxwell–Fokker–Planck system

 Access to full text

 Full (PDF)

1. [1] Arnold A., Carrillo J.-A., Gamba I., Shu C.-W., Low and high field scaling limits for the Vlasov– and Wigner–Poisson–Fokker–Planck system, Transport Theory Statist. Phys.30 (2–3) (2001) 121-153. Zbl1106.82381MR1848592
2. [2] Arnold A., Markowich P., Toscani G., Unterreiter A., On convex Sobolev inequalities and the rate of convergence to equilibrium for Fokker–Planck type equations, Comm. Partial Differential Equations26 (1–2) (2001) 43-100. Zbl0982.35113MR1842428
3. [3] Bakry D., Emery M., Hypercontractivité de semi-groupes de diffusion, C. R. Acad. Sci. Paris Sér. I Math.299 (15) (1984) 775-778. Zbl0563.60068MR772092
4. [4] Bardos C., Golse F., Levermore C.D., Fluid dynamic limits of kinetic equations II. Convergence proofs for the Boltzmann equation, Comm. Pure Appl. Math.XLVI (1993) 667-753. Zbl0817.76002MR1213991
5. [5] Ben Abdallah N., Degond P., Markowich P., Schmeiser C., High field approximation of the spherical harmonics expansion model for semiconductors, Z. Angew. Math. Phys.52 (2) (2001) 201-230. Zbl1174.82345MR1834528
6. [6] Bers A., Delcroix J.-L., Physique des plasmas, EDP Sciences, 2000. 
7. [7] Berthelin F., Vasseur A., From kinetic equations to multidimensional isentropic gas dynamics before shocks, SIAM J. Math. Anal.36 (6) (2005) 1807-1835. Zbl1130.35090MR2178222
8. [8] Bostan M., Goudon Th., Low field regime for the relativistic Vlasov–Maxwell–Fokker–Planck system; the one and one half dimensional case, Kinetic Related Models1 (1) (2008) 139-169. Zbl1185.35287MR2383720
9. [9] Bouchut F., Existence and uniqueness of a global smooth solution for the Vlasov–Poisson–Fokker–Planck system in three dimensions, J. Funct. Anal.111 (1993) 239-258. Zbl0777.35059MR1200643
10. [10] Bouchut F., Smoothing effect for the nonlinear Vlasov–Poisson–Fokker–Planck system, J. Differential Equations122 (1995) 225-238. Zbl0840.35053MR1355890
11. [11] Bouchut F., Golse F., Pallard C., Classical solutions and the Glassey–Strauss theorem for the 3D Vlasov–Maxwell system, Arch. Ration. Mech. Anal.170 (1) (2003) 1-15. Zbl1044.76075MR2012645
12. [12] Bourbaki N., Éléments de Mathématiques, Fascicule XXXV, Livre VI, Chapitre IX, Intégration, Hermann, Paris, 1969. Zbl0026.38902
13. [13] Brenier Y., Convergence of the Vlasov–Poisson system to the incompressible Euler equations, Comm. Partial Differential Equations25 (2000) 737-754. Zbl0970.35110MR1748352
14. [14] Brenier Y., Mauser N., Puel M., Incompressible Euler and e-MHD as scaling limits of the Vlasov–Maxwell system, Commun. Math. Sci.1 (3) (2003) 437-447. Zbl1089.35048MR2069939
15. [15] Carrillo J.-A., Soler J., On the initial value problem for the Vlasov–Poisson–Fokker–Planck system with initial data in $L^p$ spaces, Math. Methods Appl. Sci.18 (10) (1995) 825-839. Zbl0829.35096MR1343393
16. [16] Carrillo J.-A., Labrunie S., Global solutions for the one-dimensional Vlasov–Maxwell system for laser-plasma interaction, Math. Models Methods Appl. Sci.16 (1) (2006) 19-57. Zbl1106.35110MR2194980
17. [17] Cercignani C., Gamba I.M., Levermore C.D., High field approximations to a Boltzmann–Poisson system and boundary conditions in a semiconductor, Appl. Math. Lett.10 (4) (1997) 111-117. Zbl0894.76072MR1458163
18. [18] Chandrasekhar S., Brownian motion, dynamical friction and stellar dynamics, Rev. Mod. Phys.21 (1949) 383-388. Zbl0036.43003MR31822
19. [19] Csiszar I., Information-type measures of difference of probability distributions and indirect observations, Studia Sci. Math. Hungar.2 (1967) 299-318. Zbl0157.25802MR219345
20. [20] Degond P., Global existence of smooth solutions for the Vlasov–Fokker–Planck equations in 1 and 2 space dimensions, Ann. Scient. Ecole Normale Sup.19 (1986) 519-542. Zbl0619.35087MR875086
21. [21] Degond P., Jungel A., High field approximation of the energy-transport model for semiconductors with non-parabolic band structure, Z. Angew. Math. Phys.52 (6) (2001) 1053-1070. Zbl0991.35043MR1877692
22. [22] DiPerna R., Lions P.-L., Global weak solutions of Vlasov–Maxwell systems, Comm. Pure Appl. Math.42 (6) (1989) 729-757. Zbl0698.35128MR1003433
23. [23] B. Dubroca, R. Duclous, F. Filbet, V. Tikhonchuk, High order resolution of the Maxwell–Fokker–Planck–Landau model intended for ICF/Fast ignition applications, CELIA-Université Bordeaux 1, in preparation. Zbl1221.78051
24. [24] Glassey R., Strauss W., Singularity formation in a collisionless plasma could occur only at high velocities, Arch. Ration. Mech. Anal.9 (1) (1986) 59-90. Zbl0595.35072MR816621
25. [25] Golse F., Saint-Raymond L., The Vlasov–Poisson system with strong magnetic field in quasineutral regime, Math. Models Methods Appl. Sci.13 (5) (2003) 661-714. Zbl1053.82032MR1978931
26. [26] Goudon T., Nieto J., Poupaud F., Soler J., Multidimensional high-field limit of the electrostatic Vlasov–Poisson–Fokker–Planck system, J. Differential Equations213 (2) (2005) 418-442. Zbl1072.35176MR2142374
27. [27] Goudon T., Hydrodynamic limit for the Vlasov–Poisson–Fokker–Planck system: Analysis of the two-dimensional case, Math. Models Methods Appl. Sci.15 (5) (2005) 737-752. Zbl1074.82021MR2139941
28. [28] Goudon T., Jabin P.-E., Vasseur A., Hydrodynamic limits for the Vlasov–Navier–Stokes equations. Part II: Fine particles regime, Indiana Univ. Math. J.53 (2004) 1517-1536. Zbl1085.35117MR2106334
29. [29] Guo Y., The Vlasov–Maxwell–Boltzmann system near Maxwellians, Invent. Math.153 (3) (2003) 593-630. Zbl1029.82034MR2000470
30. [30] V. Grandgirard, Y. Sarrazin, X. Garbet, G. Dif-Pradalier, P. Ghendrih, N. Crouseilles, G. Latu, E. Sonnendrucker, N. Besse, P. Bertrand, GYSELA, a full-f global gyrokinetic semi-Lagrangian code for ITG turbulence simulations, in: Proceedings of Theory of Fusion Plasmas, Varenna, 2006. Zbl1129.35462
31. [31] Klainerman S., Staffilani G., A new approach to study the Vlasov–Maxwell system, Comm. Pure Appl. Anal.1 (1) (2002) 103-125. Zbl1037.35088MR1877669
32. [32] Kullback S., A lower bound for discrimination information in terms of variation, IEEE Trans. Inform. Theory4 (1967) 126-127. 
33. [33] Lai R., On the one and one-half dimensional relativistic Vlasov–Maxwell–Fokker–Planck system with non-vanishing viscosity, Math. Meth. Appl. Sci.21 (1998) 1287-1296. Zbl0911.35091MR1642550
34. [34] Markowich P., Ringhofer C., Quantum hydrodynamics for semiconductors in the high field case, Appl. Math. Lett.7 (5) (1994) 37-41. Zbl0814.35128MR1350607
35. [35] Nieto J., Poupaud F., Soler J., High-field limit of the Vlasov–Poisson–Fokker–Planck system, Arch. Ration. Mech. Anal.158 (2001) 29-59. Zbl1038.82068MR1834113
36. [36] O'Dwyer B., Victory H.D., On classical solutions of the Vlasov–Poisson–Fokker–Planck system, Indiana Univ. Math. J.39 (1) (1990) 105-156. Zbl0674.60097MR1052014
37. [37] Poupaud F., Runaway phenomena and fluid approximation under high fields in semiconductors kinetic theory, Z. Angew. Math. Mech.72 (1992) 359-372. Zbl0785.76067MR1178932
38. [38] Poupaud F., Soler J., Parabolic limit and stability of the Vlasov–Poisson–Fokker–Planck system, Math. Models Methods Appl. Sci.10 (7) (2000) 1027-1045. Zbl1018.76048MR1780148
39. [39] Puel M., Saint-Raymond L., Quasineutral limit for the relativistic Vlasov–Maxwell system, Asymptotic Anal.40 (2004) 303-352. Zbl1072.35181MR2107635
40. [40] Saint-Raymond L., Convergence of solutions to the Boltzmann equation in the incompressible Euler limit, Arch. Ration. Mech. Anal.166 (2003) 47-80. Zbl1016.76071MR1952079
41. [41] Vasseur A., Recent results on hydrodynamic limits, in: Dafermos C.M., Pokorny M. (Eds.), Handbook of Differential Equations: Evolutionary Equations, vol. 4, Elsevier, 2008. Zbl1185.35003MR2508169
42. [42] Victory H.D., On the existence of global weak solutions for the Vlasov–Poisson–Fokker–Planck system, J. Math. Anal. Appl.160 (2) (1991) 525-555. Zbl0764.35024MR1126136
43. [43] Wollman S., An existence and uniqueness theorem for the Vlasov–Maxwell system, Comm. Pure Appl. Math.37 (1984) 457-462. Zbl0592.45010MR745326
44. [44] Yau H.T., Relative entropy and hydrodynamics of Ginzburg–Landau models, Lett. Math. Phys.22 (1) (1991) 63-80. Zbl0725.60120MR1121850
45. [45] Yu H., Global classical solution of the Vlasov–Maxwell–Landau system near Maxwellians, J. Math. Phys.45 (11) (2004) 4360-4376. Zbl1064.82035MR2098143